El e c t ro n ic
Jo ur
n a l o
f P
r o
b a b il i t y
Vol. 15 (2010), Paper no. 68, pages 2069–2086. Journal URL
http://www.math.washington.edu/~ejpecp/
Standard Spectral Dimension for the Polynomial
Lower Tail Random Conductances Model
Omar Boukhadra
∗Université de Provence† et Université de Constantine‡ Abstract
We study models of continuous-time, symmetric, Zd-valued random walks in random environment, driven by a field of i.i.d. random nearest-neighbor conductancesωx y ∈
[0, 1]with a power law with an exponentγnear 0. We are interested in estimating the quenched asymptotic behavior of the on-diagonal heat-kernelhω
t (0, 0). We show that for
γ >d
2, the spectral dimension is standard, i.e.,
−2 lim t→+∞logh
ω
t(0, 0)/logt=d.
As an expected consequence, the same result holds for the discrete-time case.
Key words: Markov chains, Random walk, Random environments, Random conduc-tances, Percolation.
AMS 2000 Subject Classification:Primary 60G50; 60J10; 60K37.
Submitted to EJP on January 6, 2010, final version accepted November 26, 2010.
∗To my father Youcef Bey
†Centre de Mathématiques et Informatique (CMI), Université de Provence, 39 rue F. Joliot-Curie 13453
Mar-seille cedex 13, France.boukhadra@cmi.univ-mrs.fr
‡Département de Mathématiques, Université de Constantine, BP 325, route Ain El Bey, 25017, Constantine,
1.INTRODUCTION
We study the model of random walk among polynomial lower tail random conductances on
Zd, d ≥2. Our aim is to derive estimates on the asymptotic behavior of the heat-kernel in
the absence of uniform ellipticity assumption. This paper follows up recent results of Fontes
and Mathieu[9], Berger, Biskup, Hoffman and Kozma[3], and Boukhadra[5].
1.1 Describing the model.
Let us now describe the model more precisely. We consider a family of symmetric, irreducible,
nearest-neighbors Markov chains taking their values in Zd, d ≥ 2, and constructed in the
following way. LetΩbe the set of functionsω:Zd×Zd→R+such thatω(x,y) =ωx y >0
iff x ∼ y, and ωx y = ωx y ( x ∼ y means that x and y are nearest-neighbors). We call
elements ofΩenvironments.
Define the transition matrix
Pω(x,y) = ωx y
πω(x)
, (1.1)
and the associated Markov generator
(Lωf)(x) =X
y∼x
Pω(x,y)[f(y)− f(x)]. (1.2)
X = {X(t),t ∈ R+} will be the coordinate process on path space (Zd)R+ and we use the
notation Pxωto denote the unique probability measure on path space under which X is the
Markov process generated by (1.2) and satisfying X(0) = x, with expectation henceforth
denoted byEωx . This process can be described as follows. The moves are those of the discrete
time Markov chain with transition matrix given in (1.1) started atx, but the jumps occur after
independent Poisson (1) waiting times. Thus, the probability that there have been exactlyn
jumps at time t ise−ttn/n! and the probability to be at y after exactly njumps at time t is
e−ttnPωn(x,y)/n!.
Sinceωx y >0 for all neighboring pairs(x,y),X(t)is irreducible under the “quenched law Pxω" for all x. The sum πω(x) =
P
yωx y defines an invariant, reversible measure for the
corresponding (discrete) continuous-time Markov chain.
The continuous time semigroup associated withLωis defined by
Define the heat-kernel, that is the kernel ofPtωwith respect to πω, or the transition density ofX(t), by
hωt(x,y):= P
ω t (x,y)
πω(y)
(1.4)
Clearly,hωt is symmetric, that ishωt (x,y) =hωt (y,x)and satisfies the Chapman-Kolmogorov equation as a consequence of the semigroup lawPtω+s=PtωPsω. We call “spectral dimension"
ofX the quantity
−2 lim
t→+∞
loghωt (x,x)
logt , (1.5)
(if this limit exists). In the discrete-time case, we inverse the indices and denote the heat-kernel byhnω(x,y).
Such walks under the additional assumptions of uniform ellipticity,
∃α >0 : Q(α < ωb<1/α) =1,
have the standard local-CLT like decay of the heat kernel as proved by Delmotte[6]:
Pωn(x,y)≤ c1
nd/2exp
¨
−c2|x−y| 2
n
«
, (1.6)
wherec1,c2are absolute constants.
Once the assumption of uniform ellipticity is relaxed, matters get more complicated. The most-intensely studied example is the simple random walk on the infinite cluster of
super-critical bond percolation onZd,d ≥2. This corresponds toωx y ∈ {0, 1}i.i.d. withQ(ωb =
1) > pc(d) where pc(d) is the percolation threshold (cf. [10]). Here an annealed
invari-ance principle has been obtained by De Masi, Ferrari, Goldstein and Wick[7, 8]in the late
1980s. More recently, Mathieu and Remy[15]proved the on-diagonal (i.e., x = y) version
of the heat-kernel upper bound (1.6)—a slightly weaker version of which was also obtained
by Heicklen and Hoffman[11]—and, soon afterwards, Barlow[1]proved the full upper and
lower bounds on Pωn(x,y)of the form (1.6). (Both these results hold fornexceeding some
random time defined relative to the environment in the vicinity of x and y). Heat-kernel
upper bounds were then used in the proofs of quenched invariance principles by Sidoravicius
and Sznitman[16]ford ≥4, and for alld ≥2 by Berger and Biskup[2]and Mathieu and
Piatnitski[14].
LetBd denote the set of (unordered) nearest-neighbor pairs inZd. We choose in our case the
familyω={ω(x,y) : x ∼ y,(x,y)∈Zd×Zd}= (ωb)b∈Bd ∈(0,∞)B
d
i.i.d according to a lawQon(R∗+)Zd such that
ωb≤1 for allb;
where γ >0 is a parameter. In general, given functions f and g, we write f ∼ g to mean that f(a)/g(a)tends to 1 whenatends to some limit a0.
Our work is motivated by the recent study of Fontes and Mathieu [9] of continuous-time
random walks onZd with conductances given by
ωx y =ω(x)∧ω(y)
for i.i.d. random variables ω(x) > 0 satisfying (1.7). For these cases, it was found that
the annealed heat-kernel,RdQ(ω)P0ω(X(t) =0), exhibits opposite behaviors,standardand anomalous, depending whetherγ≥d/2 orγ <d/2. Explicitly, from ([9], Theorem 4.3) we
have Z
dQ(ω)P0ω(X(t) =0) =t−(γ∧d2)+o(1), t→ ∞. (1.8)
Further, in a more recent paper [5], we show that the quenched heat-kernel exhibits also
opposite behaviors, anomalous and standard, for small and large values ofγ. We first prove
for alld ≥ 5 that the return probability shows an anomalous decay that approaches (up to
sub-polynomial terms) a random constant timesn−2 when we push the powerγto zero. In
contrast, we prove that the heat-kernel decay is as close as we want, in a logarithmic sense,
to the standard decay n−d/2 for large values of the parameterγ, i.e. : there exists a positive
constantδ=δ(γ)depending only ond andγsuch thatQ−a.s.,
lim sup n→+∞
sup x∈Zd
logPωn(0,x)
logn ≤ −
d
2+δ(γ) and δ(γ)−−−→γ→+∞ 0, (1.9)
These results are a follow up on a paper by Berger, Biskup, Hoffman and Kozma[3], in which
the authors proved a universal (non standard) upper bound for the return probability in a system of random walk among bounded (from above) random conductances. In the same paper, these authors supplied examples showing that their bounds are sharp. Nevertheless, the tails of the distribution near zero in these examples was very heavy.
1.2 Main results.
Consider random walk in reversible random environment defined by a family (ωb) ∈Ω =
[0, 1]Bd of i.i.d. random variables subject to the conditions given in (1.7), that we refer to as conductances, Bd being the set of unordered nearest-neighbor pairs (i.e., edges) ofZd. The
law of the environment is denoted byQ.
We are interested in estimating the decay of the quenched heat-kernelhωt (0, 0), ast tends to
the quenched case, a similar result to (1.8). Although our result is true for alld≥2, but it is
significant whend≥5.
The main result of this paper is as follows:
Theorem 1.1 For anyγ >d/2, we have
lim t→+∞
loghωt (0, 0)
logt =−
d
2, Q−a.s. (1.10)
Our arguments are based on time change, percolation estimates and spectral analysis. In-deed, one operates first a time change to bring up the fact that the random walk viewed only on a strong cluster (i.e. constituted of edges of order 1) has a standard behavior. Then, we show that the transit time of the random walk in a hole is “negligible" by bounding the trace of a Markov operator that gives us the Feynman-Kac Formula and this by estimating its spectral gap.
An expected consequence of this Theorem is the following corollary, whose proof is given in part 3.3 and that gives the same result for the discrete-time case. For the random walk
associ-ated with the transition probabilities given in (1.1) for an environmentωwith conductances
satisfying the assumption (1.7), we have
Corollary 1.2 For anyγ >d/2, we have
lim n→+∞
logh2ωn(0, 0)
logn =−
d
2, Q−a.s. (1.11)
Remark 1.3 As it has been pointed out in[4], Remark 2.2, the invariance principle (CLT)
(cf. Theorem 1.3 in[13]) and the Spatial Ergodic Theorem automatically imply the standard
lower bound on the heat-kernel under weaker conditions on the conductances. Indeed, letC
represents the set of sites that have a path to infinity along bonds with positive conductances.
Forωx y ∈[0, 1]and the conductance law is i.i.d. subject to the condition that the probability
ofωx y >0 exceeds the threshold for bond percolation onZd, we have then by the Markov
property, reversibility ofX and Cauchy-Schwarz
hωt(0, 0) ≥ 1
2d
X
x∈C |x|≤pt
P0ω(X(t/2) =x)2
≥ 1
2d
P0ω|X(t/2)| ≤pt2
|C ∩[−pt,+pt]d|
≥ c(ω)
with c(ω) > 0 a.s. on the set {0 ∈ C } and t large enough. Note that, in d = 2, 3, this
complements nicely the “universal” upper bounds derived in [3], Theorem 2.1. Thus, for
d ≥2, we have
lim inf t→+∞
loghωt (0, 0)
logt ≥ −
d
2 Q−a.s. (1.12)
and for the cases d =2, 3, 4, we have already the limit (1.11) under weaker conditions on
the conductances (see [3], Theorem 2.2). So, under assumption (1.7), it remains to study
the cases whered≥5 and prove that forγ >d/2,
lim sup t→+∞
loghωt(0, 0)
logt ≤ −
d
2 Q−a.s.
or equivalently, since the conductances areQ−a.s. positive by (1.7),
lim sup t→+∞
logPtω(X(t) =0)
logt ≤ −
d
2 Q−a.s. (1.13)
2.ATIME CHANGED PROCESS
In this section we introduce a concept that is becoming a standard fact in the fields of random walk in random environment, namely random walk on the infinite strong cluster.
Choose a threshold parameterξ >0 such thatQ(ωb≥ξ)>pc(d). The i.i.d. nature of the
measure Q ensures that for Q almost any environmentω, the percolation graph(Zd,{e∈
Bd;ωb≥ξ})has a unique infinite cluster that we denote withCξ=Cξ(ω).
We will refer to the connected components of the complement ofCξ(ω)inZd asholes. By
definition, holes are connected sub-graphs of the grid. The sites which belong toCξ(ω)are
all the endvertices of its edges. The other sites belong to the holes. Note that holes may
contain edges such thatωb≥ξ.
First, we will give some important characterization of the volume of the holes, see Lemma 5.2 in[13]. C denotes theinfinite cluster.
Lemma 2.1 There exists¯p<1such that for p>¯p, for almost any realization of bond perco-lation of parameter p and for large enough n, any connected component of the complement of the infinite clusterC that intersects the box[−n,n]d has volume smaller than(logn)5/2.
For the rest of the section, chooseξ >0 such thatQ(ωb≥ξ)>¯p.
Define the conditioned measure
Consider the following additive functional of the random walk :
Aξ(t) = Z t
0
1{X(s)∈Cξ}ds,
its inverse(Aξ)−1(t) =inf{s;Aξ(s)>t
}and define the corresponding time changed process
Xξ(t) =X((Aξ)−1(t)).
Thus the process Xξ is obtained by suppressing in the trajectory of X all the visits to the
holes. Note that, unlikeX, the processXξ may perform long jumps when straddling holes.
As X performs the random walk in the environmentω, the behavior of the random process
Xξis described in the next
Proposition 2.2 Assume that the origin belongs toCξ. Then, under P0ω, the random process Xξis a symmetric Markov process onCξ.
The Markov property, which is not difficult to prove, follows from a very general argument
about time changed Markov processes. The reversibility of Xξ is a consequence of the
re-versibility ofX itself as will be discussed after equation (2.2).
The generator of the processXξ has the form
Lξωf(x) = 1
ηω(x)
X
y
ωξ(x,y)(f(y)−f(x)), (2.1)
where
ωξ(x,y)
ηω(x) = limt→0 1
tP ω x (X
ξ(t) = y)
= Pxω(yis the next point inCξvisited by the random walk), (2.2)
if both x and y belong toCξandωξ(x,y) =0 otherwise.
The function ωξ is symmetric : ωξ(x,y) =ωξ(y,x) as follows from the reversibility of X
and formula (2.2), but it is no longer of nearest-neighbor type i.e. it might happen that
ωξ(x,y) 6= 0 although x and y are not neighbors. More precisely, one has the following
picture : ωξ(x,y) =0 unless eitherx and y are neighbors andω(x,y)≥ξ, or there exists a
hole,h, such that both x and y have neighbors inh. (Both conditions may be fulfilled by the
same pair(x,y).)
Consider a pair of neighboring pointsx and y, both of them belonging to the infinite cluster
Cξand such thatω(x,y)≥ξ, then
This simple remark will play an important role. It implies, in a sense that the parts of the
trajectory ofXξ that consist in nearest-neighbors jumps are similar to what the simple
sym-metric random walk on Cξ does. Precisely, we will need the following important fact that
Xξobeys the standard heat-kernel bound :
Lemma 2.3 There exists a constant c such thatQξ0−a.s.for large enough t, we have
P0ω(Xξ(t) = y)≤ c
td/2, ∀y∈Z
d (2.4)
For a proof, we refer to[13], Lemma 4.1. In the discrete-time case, see[3], Lemma 3.2.
3.PROOF OFTHEOREM 1.1ANDCOROLLARY1.2
The upper bound (1.13) will be discussed in part 3.2 and the proof of Corollary 1.2 is given in part 3.3. We first start with some preliminary lemmata.
3.1 Preliminaries.
First, let us recall the following standard fact from Markov chain theory :
Lemma 3.1 The function t7−→P0ω(X(t) =0)is non increasing.
Proof. This is an immediate consequence of the semigroup property of the family of bounded self-adjoint operators(Ptω)t≥0 in the space L2(Zd,πω) endowed with the inner product de-fined by
f,g
ω=
X
x∈Zd
f(x)g(x)πω(x).
Next, let Br= [−r,r]d be the box centered at the origin and of radius r that we choose as a
function of time such that t ∼ r2(logr)−b whent → ∞, with b>1, and let B
r denote the
set of nearest-neighbor bonds ofBr, i.e.,Br={b= (x,y):x,y∈Br,x ∼ y}. We have
Lemma 3.2 Under assumption(1.7),
lim r→+∞
log infb∈B
rωb
logr =−
d
Thus, for arbitraryµ >0, we can writeQ−a.s. for r large enough,
inf b∈Br
ωb≥r−
(dγ+µ)
. (3.2)
Proof. This is a restatement of Lemma 3.6 of[9].
Consider now the following formula
Rωt f(x) =Eωx
h
f(X(t))e−λAξ(t)
i
, t≥0,λ≥0, x∈Zd (3.3)
This object will play a key role in our proof of Theorem 1.1. Let L2b(Zd,πω)denote the set of bounded functions ofL2(Zd,πω). We have
Proposition 3.3 R = {Rωt,t ≥ 0} defines a semigroup (of symmetric operators) on L2b(Zd,πω), with generator
Gωf =Lωf −λϕf (3.4)
whereϕ=1{· ∈Cξ}. One also has the perturbation identities
Rωt f(x) =Ptωf(x)−λ
Z t
0
Ptω(ϕRωt−sf)(x)ds
=Ptωf(x)−λ
Z t
0
Rωs (ϕPtω−sf)(x)ds,
t ≥0,x ∈Zd,f ∈L2b(Zd,πω).
(3.5)
Proof. The proof, that we give here, very closely mimics the arguments of[17], Theorem 1.1.
Indeed, we begin with the proof of (3.5). Observe that Pxω−a.s., for every x ∈Zd, the time
functiont7→e−λAξ(t)is continuous, and
lim h→0
1 h
e−λAξ(s+h)−e−λAξ(s)
=−λϕ(X(s))e−λAξ(s), ∀s∈[0,t], t>0,
except possibly for a countable set{αi}i∈I⊂(0,t],|I| ⊂N∗. Then, fort ≥0, we have
e−λAξ(t) = 1−λ
Z t
0
ϕ(X(s))e−λAξ(s)ds
= 1−λ
Z t
0
ϕ(X(s))exp
¨ −
Z t
s
λϕ(X(u))du
«
Multiplying both sides of the first equality of (3.6) by f(X(t))and integrating we find :
which is the first identity of (3.5). Analogously we find the second identity of (3.5) with the help of the second line of (3.6). This completes the proof of (3.5).
Clearly||Rωt f||22≤c(t,ϕ)||Ptωf||22, thenRωt f ∈L2b(Zd,πω), for f ∈L2b(Z
semigroup follows readily by letting t tend to 0 in (3.5).
Let us finally prove (3.4). To this end notice that for f ∈L2b(Zd,πω):
r be respectively the restrictions of the operatorsL
ωand
Gω(cf. (1.2)–(3.4))
to the set of functions onBrwith Dirichlet boundary conditions outsideBr, that we denote by
L2(Br,πω)(that is,LrωandG ω
r are respectively the generators of the processX and of the
semigroupR, which coincide with the ones given byLωandGωuntil the processX leaves
Br for the first time, and then it is killed). Then −Lrω and −Grω are positive symmetric
operators and we have
Grωf =L ω
r f −λϕf,
with associated semigroup defined by
(Rω,t rf)(0):=E0ω
h
f(X(t))e−λAξ(t);t< τr
i
whereτr is the exit time for the processX from the boxBr.
Let{λωi (r),i∈[1, #Br]}be the set of eigenvalues of−Grω labelled in increasing order, and
{ψω,i r,i∈[1, #Br]}the corresponding eigenfunctions with due normalization inL2(Br,πω).
3.2 Proof of the upper bound.
In this last part, we will complete the proof of Theorem 1.1 by giving the proof of the upper bound (1.13).
Proof of Theorem 1.1.
Assume that the origin belongs toCξ. By lemma 3.1, we have
P0ω(X(t) =0)≤ 2
The additive functionalAξbeing a continuous increasing function of the time, so by operating
a variable change by settings= (Aξ)−1(u)(i.e. u=Aξ(s)), we get
which is bounded by
and using lemma 2.3,
It remains to estimate the second term in the right-hand side of the last inequality, i.e. P0ω(Aξ(t/2) ≤ tε) or more simply P0ω(Aξ(t) ≤ 2ǫtε), but we can neglect the constant 2ε in the calculus as one will see in (3.15).
For eachλ≥0, Chebychev inequality gives
P0ω(Aξ(t)≤tε) = P0ω(Aξ(t)≤tε;t< τr) +P0ω(A
From the Carne-Varopoulos inequality, it follows that
P0ω(τr≤t)≤C t rd−1e−
r2
4t +e−c t, (3.9)
where C andc are numerical constants, see Appendix C in[15]. With our choice ofr such
that t∼ r2(logr)−b (b>1), we get thatP0ω(τr ≤t)decays faster than any polynomial ast
tends to+∞.
Thus Theorem 1.1 will be proved if we can check, for a particular choice ofλ >0 that may
depend ont, that
lim sup
decays faster than any polynomial intasttends
to+∞.
The Dirichlet form of −Lrω on L
2(B
r,πω) endowed with the usual scalar product (see
Lemma 3.1), can be written as
Eω,r(f,f) =¬−Lrωf,f¶ω= 1
2
X
b∈Br+1
where d f(b) = f(y)− f(x) and the sum ranges over b = (x,y) ∈ Brω+1. By the min-max
Theorem (see[12]) and (3.4), we have
λω1(r) = inf f6≡0
Eω,r(f,f) +λPx∈Cξ
r f
2(x)π ω(x)
πω(f2)
. (3.11)
where Crξ is the largest connected component of Cξ∩Br, and the infimum is taken over
functions with Dirichlet boundary conditions. (Recall that λω1(r) is the first eigenvalue of
−Grω.)
To estimate the decay of the first term in the right-hand side of (3.8), we will also need to estimate the first eigenvalueλω1(r). Recall thatµdenotes an arbitrary positive constant.
Lemma 3.4 Under assumption(1.7), for any d ≥2andγ >0, we haveQ−a.s.for r large enough,
λω1(r)≥(8d)−1r−(
d
γ+µ)(logn)−5, (3.12)
forλproportional to r−(
d
γ+µ).
Proof. For some arbitraryµ >0, let r be large enough so that (3.2) holds. Lethbe a hole
that intersects the boxBr, and for notational ease we will use the same notation forh∩Br.
Define ∂hto be the outer boundary ofh, i.e. the set of sites in Crξ which are adjacent to
some vertex inh. Let us associate to each holeha fixed siteh∗∈ Crξ situated at the outer
boundary ofhand for x∈hcallκ(x,h∗)a self-avoiding path included inhwith end points x
andh∗, and let|κ(x,h∗)|denote the length of such a path.
Now let f ∈L2(Br,πω)and letBω(h)denote the set of the bonds ofh. For each x∈h, write
f(x) = X
b∈κ(x,h∗)
d f(b) +f(h∗)
and, using Cauchy-Schwarz
f2(x)≤2|κ(x,h∗)| X
b∈κ(x,h∗)
|d f(b)|2+2f2(h∗).
and sum over x∈hto obtain
where in the last term, we multiply by 2dsince we may associate the sameh∗to 2d different
Let us get back to the proof of the upper bound. Let
Then, for large enough t and by (3.14), we have
eλtεE0ω
In conclusion, asµis arbitrary and according to (3.9)–(3.10), we obtain
and finally, by (3.7)
lim
ε→1lim supt→+∞
logP0ω(X(t) =0)
logt ≤ −
d
2 for γ > d 2,
which gives (1.13).
We conclude that for any sufficiently small ξ, then Qξ0−a.s. (1.11) is true and since
Q∪ξ>0{0∈ Cξ(ω)}
=1, it remains trueQ-a.s.
3.3 Proof of the discrete-time case.
Proof. In the same way, the lower bound holds by the Invariance Principle (cf. [4]) and the Spatial Ergodic Theorem (see Remark 1.3).
For the upper bound, let(Nt)t≥0 be a Poisson process of rate 1. Setn=⌊t⌋. Pω2n(0, 0)being
a non increasing function ofn(cf. Lemma 3.1), then
P0ω(X(t) =0) = e−tX k≥0
tk k!P
k ω(0, 0)
≥ e−t 2n
X
k=0 tk k!P
k ω(0, 0)
≥ Pω2n(0, 0)
e−t
2n
X
k=0 tk k!
= Pω2n(0, 0)Prob(Nt≤2n).
By virtue of the LLN, we have
Prob(Nt≤2n)−−−→
t→+∞ 1.
From here the claim follows.
A
CKNOWLEDGMENTSI would like to thank Pierre Mathieu, my Ph.D. advisor, for having suggested me the problem and for the numerous fruitful discussions about this subject. Also, I would like to thank Marek Biskup and Nina Gantert, my Ph.D. referees, who helped me correct and improve the paper.
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